THE ASYMPTOTIC BEHAVIOR AND SYMMETRY OF POSITIVE SOLUTIONS TO p-LAPLACIAN EQUATIONS IN A HALF-SPACE

We study a nonlinear equation in the half-space with a Hardy potential, specifically, -Delta(p)u = lambda u(p-1)/x(1)(p) - x(1)(theta) f(u) in T, where Delta(p) stands for the p-Laplacian operator defined by Delta(p)u = div(vertical bar del u vertical bar(p-2)del u), p > 1, theta > -p, and T is a half-space {x(1) > 0}. When lambda > Theta (where Theta is the Hardy constant), we show that under suitable conditions on f and theta, the equation has a unique positive solution. Moreover, the exact behavior of the unique positive solution as x(1) -> 0(+), and the symmetric property of the positive solution are obtained.